Self-organized criticality induced by quenched disorder experiments on flux avalanches in N

a r X i v :c o n d -m a t /0410369v 1 [c o n d -m a t .s t a t -m e c h ] 14 O c t 2004

Self-organized criticality induced by quenched disorder:experiments on ?ux

avalanches in NbH x ?lms.

M.S.Welling,C.M.Aegerter ?,and R.J.Wijngaarden

Division of Physics and Astronomy,Faculty of Sciences,Vrije Universiteit,De Boelelaan 1081,1081HV Amsterdam,The

Netherlands (February 2,2008)

We present an experimental study of the in?uence of quenched disorder on the distribution of ?ux avalanches in type-II superconductors.In the presence of much quenched disorder,the avalanche sizes are power-law distributed and show ?nite size scaling,as expected from self-organized criticality (SOC).Furthermore,the shape of the avalanches is observed to be fractal.In the absence of quenched disorder,a preferred size of avalanches is observed and avalanches are smooth.These observations indicate that a certain minimum amount of disorder is necessary for SOC behavior.We relate these ?ndings to the appearance or non-appearance of SOC in other experimental systems,particularly piles of sand.

PACS numbers:05.65.+b,74.70.Ad,64.60.Ht,74.25.Qt

Self-organized criticality (SOC)[1]has generated great interest over the last 15years mainly due to its wide range of possible applications in many non-equilibrium systems [2,3].However,progress has been hampered by the fact that clear,tell-tale signs of criticality,such as ?nite-size scaling in the distribution of avalanches,have only been observed in very few controlled experiments [4].Recently however,there have been a number of ex-perimental observations of criticality in both two-[5,6]and three dimensional systems [7–9].However,the criti-cal ingredients to obtain SOC in an experimental system still remain obscure.

Recent theoretical advancements have studied the na-ture of the criticality in SOC and made a link with phase transitions describing how a moving object comes to rest [10,11].Such absorbing state phase transitions are closely related to the roughening of an elastic membrane in a random medium [12].In these theoretical models,there needs to be an absorbing state phase transition underly-ing the process at hand in order to observe SOC.This critical state is reached by a self-organization process [13],which depends on slowly driving the system away from its state where everything is at rest.If the driv-ing is not slow enough,the relaxations to the critical point may be disturbed by the driving,such that no crit-ical state is reached [14].Too strong driving may have occurred in some of the early experiments,particularly those performed in rotating drums and where the grains were dropped from a considerable height.The absence of an underlying phase transition,however,would be more fundamental and detrimental to SOC.In the case of ab-sorbing state phase transitions,it is known that the pres-ence of quenched disorder plays an important role in the nature of the critical point,such that this may also be an important ingredient in obtaining SOC behavior.Here we present an experimental investigation of the in-?uence of disorder on the appearance or non-appearance of SOC.Therefore it is necessary to have a system,where the quenched disorder can be experimentally changed while leaving all other aspects the same,as well as having

a system which has been shown to show SOC at least in some circumstances.We study the avalanche dynamics of magnetic vortices in the type-II superconductor Niobium in the presence of Hydrogen impurities using magneto-optics.As ?rst noted by de Gennes [15],the penetration of a slowly ramped magnetic ?eld into a type-II supercon-ductor has a strong analogy to the growing of a sand pile,the archetypal example of SOC.It has been shown in the past,that these vortex avalanches are distributed accord-ing to a power-law [16]and more recently that they obey ?nite-size scaling [9].However,due to the presence of pinning in the system,studying magnetic vortices allows a detailed investigation of the in?uence of quenched dis-order on their avalanche distribution and structure.N

b is particularly suited for this purpose,as it easily takes up H impurities (which locally destroy superconductiv-ity,see below)[17],which can be added to the sample via a contact gas,thus allowing an experimental control of the amount of quenched disorder in the system.Further-more,studying magneti

c vortices has the advantage that kinetic e?ects in their dynamics,which are thought to have hampere

d som

e sand-pile experiments [5],are nat-urally absent due to the over-damped dynamics o

f the vortices [18].

The experiments described here were carried out on a Nb ?lm of a thickness of 500nm,evaporated under ultra high vacuum conditions on an ’R-plane’sapphire substrate.The ?lms were then covered with a 10nm Pd caplayer in order to facilitate the catalytic uptake of H into the ?lm [19].

The local magnetic ?ux density,B z ,just above the sample is measured using an advanced magneto-optical (MO)setup,directly yielding the local Faraday-angle in an Yttrium-Iron Garnet indicator,using a lock-in tech-nique [20].Images are taken with a charge-coupled device camera (782×582pixels)with a resolution of 3.4μm per pixel.The sample is placed in a cryo-magnet and cooled to 4.2K in zero applied ?eld.The external ?eld is then ramped from 0to 20mT in steps of 50μT,where the ?ux in the sample is relaxed for 3seconds before an MO

picture is acquired.This sequence is repeated twice for each H concentration to check for reproducibility.

The quenched disorder in the sample is increased af-ter such a sequence of two?eld sweeps by equilibrating the sample with a certain higher partial pressure of H at room temperature for one hour.The partial loading pressures used in the experiments discussed here were 80,260,1130,and1810Pa.We estimate that the H im-purities present in the as grown sample correspond to a partial loading pressure lower than roughly10Pa.Af-ter equilibration,ensuring a uniform distribution of H in the Nb?lm,the sample is cooled down again.In this cooling procedure,phase separation occurs in H-rich and H-poor regions[21],where superconductivity is sup-pressed in the H-rich phase[22].Thus the clusters of H-rich phase,which are in the order of0.1to1μm in di-ameter[21],act as e?ective pinning sites for the vortices and thereby introduce quenched disorder.

FIG.1.a)The magnetic?ux landscape in the center of the Nb?lm without added Hydrogen impurities.The mag-netic?ux density in the sample is indicated by the brightness, where white corresponds to32mT and black corresponds to 0mT(see inset).b)The?ux landscape in the same sam-ple after the application of a H pressure of1810Pa at room temperature.As can be seen,the?ux structures are much more disordered and branched.c)Magnetic?ux avalanches in the absence of H treatment,as given by the subtraction of two consecutive MO images.Note that the magni?cation and the grey-scale is di?erent from a),white corresponds to 0mT and black corresponds to3mT.The avalanches have a predominant size and are rather smooth and plume-shaped.

d)Flux avalanches after H treatment with1810Pa gas pres-sure at room temperature,again obtained from a subtraction of consecutive MO images(grey scale and magni?cation are as in c).The avalanches are more fractal and branched than in c)and do no longer have a preferred size(see text).

In this manner,we obtain a collection of magnetic?ux landscapes as shown in Fig.1a),which are then analyzed in terms of the avalanches that have occurred between time steps.In order to obtain the amount of magnetic ?ux that has been transported in one avalanche,we?rst average each image over a set of4×4pixels to reduce noise.Then two consecutive images are subtracted lead-

ing to an image like that shown in Fig.1c).As can be seen in the?gure,in each image we observe a number of avalanches.These are identi?ed individually,using a threshold of0.3mT in?B z.The amount of?ux in each of these avalanches is subsequently determined from in-tegrating the di?erence in?ux density over the area of the avalanche:

s=?Φ= ?B z dxdy.(1)

From the spatial resolution of the setup and the thresh-old level of the avalanche identi?cation,we can determine the smallest avalanches that are still resolved to contain about20Φ0,whereΦ0=h/2e is the magnetic?ux car-ried by each vortex.In order to check for?nite size scal-ing,we discard avalanches exceeding a linear extent of 200,100,and50μm from the analysis.This corresponds to decreasing the system size accordingly,as the linear extent of an avalanche cannot exceed the system size.

As can be seen in Fig.1c),when the sample has not been treated with H,the avalanches have a characteris-tic structure and area.This leads to a peak in the size distribution,which disappears as avalanches exceeding the preferred linear extent are discarded.Thus in the absence of H-induced quenched disorder,no SOC behav-ior is observed,as there is no?nite-size scaling of the avalanche distribution.This is shown in Fig.2a),where we show the scaled avalanche size distribution for the sample without H treatment.Here,the size distribution has been logarithmically binned,such that avalanche size steps in the histogram are separated by a constant fac-tor rather than a constant step width.Moreover,the histograms are scaled with sτ,which leads to horizontal lines in case the avalanches are power-law distributed.

The displaced?ux per avalanche is scaled by L?D,where L is the scale above which avalanches are discarded and

D is the fractal dimension of the avalanches.In order to

obtain the’collapse’in Fig.2a),we have usedτ=1.35 and D=2.75.The value ofτis determined from a best collapse of the data onto a horizontal line for small s.The value of D is found from the best collapse of the cut-o?values.The nice collapse of cut-o?values indicates that the avalanches in this case do have a de?nite fractal di-mension,which is close to3,as one would expect given their smooth appearance in Fig.1c).The peak that can be observed in the distribution is due to the predomi-nance of avalanches containing about～2500Φ0,which can also be inferred from Fig.1c).

Turning to the data in Fig.1d),it can be seen that in the presence of strong quenched disorder(i.e.after treat-ing the sample with1810Pa of H gas),the avalanche structure is markedly di?erent from that in the pristine case.Now,the?ux moves in a much more branched way and furthermore a preferred size of avalanches is absent.Again,we have checked this with a?nite-size scaling analysis of the avalanche size distributions,see Fig.2b).The size distribution is again logarithmically

binned and scaled in order to obtain a curve collapse.Here,in contrast to Fig.2a),there is good curve col-lapse and thus we can determine the size distribution exponent and avalanche dimension to be τ=1.07(2)and D =2.25(5),where we observe a power-law distribution over more than two orders of magnitude.

1010101010

P (s ) s

τs / L

1010

P (s ) s

s / L

FIG.2.a)The ?ux avalanche size distributions in the ab-sence of Hydrogen treatment.Data are shown for sets of avalanches not exceeding di?erent linear sizes of L =200,100,and 50μm.The di?erent distributions are vertically scaled with s τand the sizes are horizontally scaled with L ?D ,in order to obtain the usual curve collapse for ?nite size scaling.However,as avalanches of ～2500Φ0are preferred,no good curve collapse is obtained in this region.For small avalanches and avalanches of the cut-o?size,the curves do collapse,such that the fractal dimension of the avalanches can be deter-mined.b)The avalanche size distributions scaled in the same way,for experiments after H treatment at 1810Pa of the sam-ple.Here a good curve collapse is observed,with power-law scaling over more than two decades.This is clear evidence that in the presence of quenched disorder,SOC is present in the vortex avalanches.

This overall behavior of ?ux avalanches in the presence of changeable quenched disorder has also been studied in molecular dynamics simulations of a collection of vor-tices [23].This is a great advantage of the vortex system over the study of sand piles,as the interactions between

vortices are well known [18]and can thus be simulated in order to separate collective e?ects from microscopic dynamics.In their study,Olson et al.[23]have found that on increasing the pinning density,the avalanche dis-tributions change from showing a preferred size (with a power-law distribution at small sizes)to a power-law dis-tribution.Furthermore,in the case of strong pinning,the exponent τobtained in these power-law distributions is between 1and 1.4in very good agreement with our experimental values.Moreover,in molecular dynamics simulations,the movements of all vortices can be followed and it turns out that in the case of low pinning density,the avalanches form channels [23],which is highly remi-niscent of the structures seen in Fig.1c).

Let us now turn to the general implications on the ap-pearance of SOC that can be drawn from the above ob-servations.As we have shown,in the absence of quenched disorder,even the non-kinetic vortices [18]do not show ?nite-size scaling in their avalanche distribution.Fur-thermore,it has been shown previously that steel balls in a two dimensional pile can show SOC [6]when quenched noise is introduced via a ?xed random arrangement of balls at the base.These two observations together seem to indicate that kinetic e?ects may be less important than previously thought [5],while it is the presence of quenched noise that leads to the appearance of SOC.This is also corroborated by molecular dynamics simu-lations on vortices,using over-damped dynamics ?nding the same result [23].

The importance of quenched disorder can be best seen in the structure of the avalanches,which is intimately connected to the surface of a pile or ?ux landscape [7,24].As the number of pins is increased,the vortex avalanches have to accommodate to their presence.This frustra-tion leads to a much more branched and open avalanche structure.This is shown in Fig.3,where the circles show the values of D for di?erent H loading pressures.As can be seen,at low pinning density,the avalanches have a dimension close to 3,indicating a smooth structure,whereas with increasing H content,the avalanche dimen-sion decreases.Due to the more complex structure of an avalanche,a bigger range of possible avalanche sizes can be explored over the area of the sample.Thus there is no longer a preferred size of avalanches and hence SOC behavior occurs.Again,we can compare our results to those of Altshuler et al.[6],where in the absence of SOC,only a small part of the pile took part in avalanches and the pile surface overall was much smoother than in the case where SOC was observed.

In order to quantify the roughness [25]of the ?ux landscape,we determine the width, (B z (x,y )? B z (x,y ) y )2 1/2,of the ?ux landscape after subtraction of the average pro?le, B z (x,y ) y .As can be seen in Fig.3,with increasing H content,the ?ux landscape be-comes more rough as given by a fourfold increase in sur-face width.Thus as has been observed in experiments using steel balls [6]as well as in rice piles [7,26],SOC be-

havior in the ?ux landscape is coupled to the observation of a rough pile surface.

101001000

s u r f a c e w i d t h (m T )

a v a l a n c h e d i m e n s i o n D

Hydrogen gas loading pressure [Pa]

FIG.3.The fractal dimension of the avalanches (cir-cles)and the surface width (see text)of the ?ux landscape (squares)as a function of Hydrogen partial pressure.Data for the pristine ?lms are shown at 10Pa,the estimated max-imum contamination H pressure.The fractal dimension de-creases with increasing quenched disorder,corresponding to a more branched and rough structure,as seen in Fig.1d).The surface width increases with quenched disorder,which indi-cates that the ?ux landscape becomes increasingly rough as H is introduced into the sample,as can be seen in Fig.1b).

In conclusion,we have shown in the case of high pin-ning density that the magnetic ?ux avalanches observed in a Nb ?lm show ?nite-size scaling,which implies the presence of SOC in the system.However,in the presence of a low pinning density,a preferred size of avalanches is found and no ?nite-size scaling is observed.This demon-strates the importance of quenched disorder in the system in order to obtain SOC behavior.The exact position of the transition from non-SOC to SOC behavior cannot be determined very accurately due to experimental limita-tions (e.g.’noise’in Fig.2and limited accessible range of L ),but might also be intrinsically smooth if one considers the gradual changes observed in Fig.3for the avalanche dimension D and surface width.Our experimental ob-servation of this transition as a function of disorder is in agreement with previous numerical work on cellular automata [27]and with molecular dynamics simulations of magnetic vortices [23].In addition,our ?ndings are consistent with the point of view of SOC as an absorbing state phase transition [10,13].In absorbing state phase transitions,such as directed percolation [25],the density of quenched disorder is a critical parameter,whereby a phase transition can be induced.The presence of such an underlying phase transition is a necessary condition in order to obtain SOC in the model of Vespigniani et al.[13].Thus again,this view advocates that the increase of quenched noise can lead to the appearance of SOC,as indeed we have found experimentally.

We would like to thank Ruud Westerwaal for help with the preparation of the samples.This work was supported by FOM (Stichting voor Fundamenteel Onderzoek der

Materie),which is ?nancially supported by NWO (Ned-erlandse Organisatie voor Wetenschappelijk Onderzoek).